MTE 280: Number Operations and Proportional Reasoning k-8 Teaching

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Turing machine vs. V. Neuman architecture

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Fractions, Decimals, and Percentages

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The Computer System

Ved Eros Maugeri

Biznes Plan

Ved Janusz Krawczak

MTE 280: Number Operations and Proportional Reasoning k-8 Teaching

Week 16

Test 3

Review

Week 14

Problem Solving W/Percentages

Problem Solving With Percentages

3 Types of Problems with %'s:

1. 8 is what percent of 22? → answer will be a %
2. 8% of 22 is what #? → answer will be #
3. 8% of what # is 22?

How to Solve:

1. 8 = n x 22 → n = 8/22 → n = 36.6^- → n = 37%
2. 0.08 x 22 = n → n = 1.76
3. 0.08 x n = 22 → n = 22/0.08 → n = 275

Essential Vocab:

☆ "is" → = ☆ Change % into decimals

☆ "of" → x ☆ "what percent" → write decimal as a percent

☆ "what" → variable n

Percentages

Percentages

☆ % = percent

☆ percent → per - cent → literally means out of 100

• 25% = 25 out of 100
• 3/100=0.03=3%
• 43/100=0.43=43%

☆ 100 does not always = 100%

• 9 people in a class everyone in there → 9=100%
• without context of the word problem 9=900%

☆ can be > 100 → 1.25=125%

Week 15

Integers

Integers

☆ Integers: positive and negative numbers

☆ "Chip Method":

• Best method for teaching integers
• Can be used for many lessons
• Uses two-color tokens to represent positive/negative numbers
• 🟡 = Positive and 🔴 = Negative
• 🟡🟡🟡 = 3 and 🔴🔴🔴 = -3

Division

Division W/Integers:

☆ Multiplication + Division = Inverse Operations

Multiplication

Multiplication W/Integers:

☆ Commutative: a x b = b x a

• 3 x 2 = 6
• 3 groups of 2 = 6
• (++) (++) (++) = 6
• repeated addition → 2+2+2

• Commutative [property of multiplication: axb=bxa
• 3 groups of 2 = 6 → (++) (++) (++) = 6
• 3 groups of ^-2 = 6 → (--) (--) (--) = ^-6
• ^-3 groups of 2 (^-3x2) = ^-6 (same because of commutative property)
• 3 x ^-2 = ^-3 x 2 = ^-6 ⤴

^{-Means "the opposite of" ⤵}

^-3 x ^-2 = the opposite of 3 x ^-2 = ^-6 → opposite of → = 6 → ^-3 x ^-2 = 6

🚫 ^-3 groupssomethinghing ⤴

Solve it this way because it is easier for students to understand ⤴

a) 2 x 5 = 2 groups of 5 = (+++++) (+++++) = 10

b) 2 x ^-5 = 2 groups of ^-5 = (-----) (-----) = ^-10

c) ^-2 x 5 = 5 x ^-2 = 5 groups of ^-2 = (--) (--) (--) (--) (--)

d) ^-2 x ^-5 = the opposite of 2 x ^-5 = 2 groups of ^-5 = (-----) (-----) = ^-10 → opposite = 10

Subtraction

Subtraction W/Integers:

🟡 = Positive and 🔴 = Negative

☆ negative +/- positive = zero pair → 🔴 🟡 = 0 → cancel eachother out

• ⁺5 - ⁺1 = ⁺4

🟡🟡🟡🟡🟡 - 🟡 = 🟡🟡🟡🟡

• ^-5 - ^-1 = ^-4

🔴🔴🔴🔴🔴 - 🔴 = 🔴🔴🔴🔴

• ⁺5 - ^-1 = ⁺6

🟡🟡🟡🟡🟡 - 🔴 → 🟡🟡🟡🟡🟡 - 🔴 (+ 🔴 🟡) → 🟡🟡🟡🟡🟡🟡🔴 - 🔴 = 🟡🟡🟡🟡🟡🟡

• ^-5 + ⁺1 = ^-6

🔴🔴🔴🔴🔴 + 🟡 → 🔴🔴🔴🔴🔴 + 🟡 (+ 🔴 🟡) → 🔴🔴🔴🔴🔴🔴🟡 - 🟡 = 🔴🔴🔴🔴🔴🔴

a) 3 - ^-1 = 4

🟡🟡🟡 - 🔴 = 🟡🟡🟡🟡

🟡🟡🟡🟡🔴 - 🔴 = 🟡🟡🟡🟡

b) ^-6 - ^-2 = ^-4

🔴🔴🔴🔴🔴🔴 - 🔴🔴 = 🔴🔴🔴🔴

c) 3 - ^-4 = 7

🟡🟡🟡 - 🔴🔴🔴🔴 = 🟡🟡🟡🟡🟡🟡🟡

🟡🟡🟡🟡🟡🟡🟡🔴🔴🔴🔴 - 🔴🔴🔴🔴 = 🟡🟡🟡🟡🟡🟡🟡

d) ^-2 - ^-3 = 1

🔴🔴 - 🔴🔴🔴 = 🟡

🔴🔴🔴🟡 - 🔴🔴🔴 = 🟡

Adition

Addition W/Integers:

🟡 = Positive and 🔴 = Negative

• ⁺5 + ⁺1= ⁺6

🟡🟡🟡🟡🟡 + 🟡 = 🟡🟡🟡🟡🟡🟡

• ^-5 + ^-1= ^-6

🔴🔴🔴🔴🔴 + 🔴 = 🔴🔴🔴🔴🔴🔴

• ⁺5 + ^-1= ⁺4

🟡🟡🟡🟡🟡 + 🔴 = 🟡🟡🟡🟡

• ^-5 + ⁺1= ^-4

🔴🔴🔴🔴🔴 + 🟡 = 🔴🔴🔴🔴

☆ negative +/- positive = zero pair → 🔴 🟡 = 0 → cancel eachother out

a) 3 + ^-2 = 1

🟡🟡🟡 + 🔴🔴 = 🟡

or 🟡🟡🟡 = 1

+ 🔴🔴

b) ^-6 + ^-2 = ^-8

🔴🔴🔴🔴🔴🔴 + 🔴🔴 = 🔴🔴🔴🔴🔴🔴🔴🔴

c) 3 + ^-4 = ^-1

🟡🟡🟡 + 🔴🔴🔴🔴 = 🔴

d) ^-2 + ^-1 = ^-3

🔴🔴 + 🔴 = 🔴🔴🔴

Week 13

Video

Decimals

Decimals

☆ Base-10 system = Decimal System = 1-to-10 Relationship

☆ Decimal separates the part form the whole

☆ Decimal sits to the right of the unit

Converting

Converting

1. Denominator is a multiple of 10
2. $7/10=0.7$
3. $12/100=0.12$
4. $3/100=0.03$
5. $1.32=1 32/100$
6. $0.72=72/100$
7. Denominator is a factor of 10, 100, 1000, ect.
8. $2/5=(2x2)/(5x2)=4/10=0.4$
9. $7/20=(7x5)/(20x5)=35/100=0.35$
10. Denominator is not 1 or 2

Repeating VS Terminating

☆ is not a decimal because it is not terminating or repeating - it is an irrational #



4 Operations

Addition:

Subtraction:

Multiplication:

Division: